Asymptotic Uncorrelation for Mexican Needlets

We recall Mexican needlets from [5]. We derive an estimate for certain types of Legendre series, which we apply to the statistical properties of Mexican needlets. More precisely, we shall show that, under isotropy and Gaussianity assumptions, the Mexican needlet coefficients of a random field on the sphere are asymptotically uncorrelated, as the frequency parameter goes to infinity. This property is important in the analysis of cosmic microwave background radiation.Comment: 13 page

Homogeneous Besov spaces on stratified Lie groups and their wavelet characterization

We establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, both in terms of continuous and discrete wavelet systems. We first introduce a notion of homogeneous Besov space $\dot{B}_{p,q}^s$ in terms of a Littlewood-Paley-type decomposition, in analogy to the well-known characterization of the Euclidean case. Such decompositions can be defined via the spectral measure of a suitably chosen sub-Laplacian. We prove that the scale of Besov spaces is independent of the precise choice of Littlewood-Paley decomposition. In particular, different sub-Laplacians yield the same Besov spaces. We then turn to wavelet characterizations, first via continuous wavelet transforms (which can be viewed as continuous-scale Littlewood-Paley decompositions), then via discretely indexed systems. We prove the existence of wavelet frames and associated atomic decomposition formulas for all homogeneous Besov spaces ${\dot B}_{p,q}^{s}$, with $1 \le p,q < \infty$ and $s \in \mathbb{R}$.Comment: 39 pages. This paper is to appear in Journal of Function Spaces and Applications. arXiv admin note: substantial text overlap with arXiv:1008.451